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Environmental Science Nano PAPER View Article Online View Journal Cite this: DOI: 10.1039/c6en00061d Reactivity of ferrihydrite and ferritin in relation to surface structure, size, and nanoparticle formation studied for phosphate and arsenate Received 4th March 2016, Accepted 23rd August 2016 DOI: 10.1039/c6en00061d rsc.li/es-nano Tjisse Hiemstra* a and Wei Zhao b Ferritin (Ftn) is a natural protein that can store metal (hydr)oxide nanoparticles of tunable size in its cavity and bind oxyanions. This quality can be used in water purification by applying nanotechnology. As our study suggests, the adsorption behavior of engineered ferritin strongly resembles that of ferrihydrite (Fh) which is in nature the most insoluble and stable Fe (hydr)oxide at the nanoscale due to its exceptionally low surface Gibbs free energy. For freshly prepared Fh, the adsorption of phosphate has been measured and modeled as a function of ph (4 11), solution concentration (μm mm), and background electrolyte level (0.01 1 M). In connection to surface complexation modeling (SCM) the surface structure of Fh has been analyzed, quantifying the unique polyhedral surface composition with singly coordinated surface groups that may form either monodentate or bidentate surface complexes. The spherical double layer theory has been implemented for downscaling to very small cluster sizes, which is vital for studying nanoparticle behavior in natural as well as synthetic systems. Modeling suggests the formation of very small Fh clusters when particles are produced in the absence of Ftn by instantaneous FeĲIII) neutralization, leading at ph 7 on average to 62 ± 3 Fe per particle with a solubility product of log Q so = 37.3 ± 0.2. The corresponding size of the critical nucleus (n Fe =11± 2) matches well with the smallest imaginable Fh cluster with a Fe 13 Keggin structure. Application of the model to engineered ferritin (Ftn) suggests that in the initial step of preparation, more than one Fe (hydr)oxide cluster per ferritin can be formed, having a very small size (n Fe 17). The initial clusters, metastable in Ftn, can grow at stepwise additions of Fe and will fuse to a large singlet above 1000 Fe per Ftn. However, the clusters remain very small ( 20 ± 5 Fe) when Ftn is loaded with Fe in the presence of PO 4. These ultra-small clusters are stabilized by the adsorption of PO 4 ( 5 10 PO 4 per cluster) that decreases the surface Gibbs free energy (G surf ). At a large decrease in G surf due to ion adsorption, Fe-oxyanion networks will form that can be important in terrestrial and aquatic systems. Nano impact Oxyanion adsorption is hardly understood for Fe (hydr)oxides at the ultra-nanometer scale between nuclei and crystallites, while this knowledge is vital for elucidating pathways of nanoparticle formation and related sequestration of elements in natural environments and technological settings. An ion adsorption model for ferrihydrite and engineered ferritin has been developed and calibrated with data. The model enables downscaling to very small cluster sizes, allowing investigation of the nucleation and nanoparticle formation in complex aqueous media. Our study quantifies the pathway of the formation of Fe (hydr)oxide in the absence and presence of oxyanions as it may proceed via formation of ultra-small nuclei that match Fe 13 Keggin clusters in size. Above a threshold value, iron-oxyanion networks may form. a Department of Soil Quality, Wageningen University, P.O. Box 47, 6700 AA Wageningen, The Netherlands. E-mail: tjisse.hiemstra@wur.nl; Fax: +31 317 41 9000; Tel: +31 317 48 2342 b State Key Laboratory of Soil Erosion and Dryland Farming on the Loess Plateau, Northwest A & F University, Yangling, Shaanxi Province 712100, PR China Electronic supplementary information (ESI) available: Electrical double layer structure, modeling primary charge, MO/DFT computations, additional phosphate adsorption data for ferrihydrite, arsenate adsorption by ferrihydrite, kinetics of PO 4 binding to Pf Ftn and Fh, PO 4 adsorption isotherms of flash-produced Fh, pathway of particle formation at homogeneous nucleation, specific surface area of Fh, arsenate adsorption of ferritin. See DOI: 10.1039/c6en00061d On leave at Wageningen University. Introduction Ferritin (Ftn) is an iron storage protein that is universally found in prokaryotes and eukaryotes controlling the bioavailability of Fe. 1 By self-assembly, subunits of the protein form a nearly spherical nano-sized cage that is able to store iron (hydr)oxide as a nanoparticle. 2 Prokaryotic ferritin is a 24- mer ferritin with an outer diameter of about 12 nm and an inner diameter of about 8 nm. The biomineral in the core of this Ftn has a maximum size of 5 6 nm. 3 In mini-ferritin,

the cage is built from 12 subunits with an external diameter of 8 nm and an internal diameter of about 6 nm. 1 In nanotechnology, ferritin cages can be used as templates for synthesizing metal and metal (hydr)oxide nanoparticles. 4 The required ferritin can be produced in large quantities by modified E. coli. 5 and loaded with different amounts of for instance Fe (hydr)oxide. 6 It allows manufacturing of nanoparticles of tunable size. In this process, the engineered nanoparticle can be built by stepwise additions of FeĲII) with intermediate enzymatic oxidation by exposure to air. 6 A small part of the added Fe will be bound in reactive centers of the protein, 2 i.e. 48 Fe per mole prokaryotic (24-mer) ferritin. The Fe in excess will form the biomineral in the cage. Iron (hydr)oxide in the cage of ferritin (Ftn) is able to bind oxyanions such as phosphate, 7 arsenate, vanadate, chromate, and other oxyanions. 8 Heavy metal ions (Ni, Cd, Cu, Co, Zn, and others) can also be incorporated. The iron (hydr)oxide nanoparticle is also photo- and redox-reactive. 9 These properties are used to produce functional materials for nano-catalysis, nano-electronics, magnetic resonance imaging, and targeted drug delivery. 2,3 An interesting application of engineered ferritin is the prevention of biofouling of membranes for drinking water production. 3 Ferritin is able to reduce the phosphate concentration to pm levels that avert microbial growth on membrane surfaces. 6 The amount of oxyanions bound by the iron (hydr)oxide core depends on the solution conditions and the particle size. 6 The PO 4 /Fe ratio of the core can be influenced by the pathway of production. In organisms, the PO 4 /Fe ratio reflects the local phosphate concentration. 3 In plants, the PO 4 / Fe is high while in humans, it is low. In vitro, iron and phosphate can be added to apo-ferritin either simultaneously or sequentially which leads to different particles. 10 The crystal structure of Fe (hydr)oxide in the cage of ferritin 9,11,12 may very well resemble the structure of ferrihydrite, 13 which is one of the most important hydrated Fe-oxide minerals in the natural environment. 14,15 Using conventional methods, it is extremely difficult to derive the structure of ferrihydrite due to the very small size and low crystal symmetry. Based on high-energy X-ray total scattering (HEXS) data, a new crystal structure has been proposed for ferrihydrite, 13 replacing a former model. 16 The polyhedral composition of the new ferrihydrite structure has been found to be particle-size dependent. 17 This important feature can be understood from the extremely large surface-to-volume ratio of these nanoparticles. 18 For freshly prepared two-line ferrihydrite (2LFh), about half of all the iron is part of the surface. The surface of ferrihydrite has a unique polyhedral and chemical composition that is different from the composition of the mineral core. 18 Specific types of structural Fe polyhedra are less well represented at the surface, i.e. the surface is depleted by certain types of polyhedra, which is further supported by recent spectroscopic information. 19 The surface depletion (SD) model 18 explains a large number of microscopic and macroscopic observations. Due to the variable contribution of the surface, Fh has a size-dependent mass density and molar mass 17,20 as well as water content. 17,18,21,22 The mineral core of ferrihydrite is nearly an oxide (Fe 10 O 14 (OH) 2 ), but the particle as a whole is water rich due to a large contribution of coordinated OH 2 and OH surface groups leading overall to an equivalent chemical composition that is typically between FeOOH and Fe(OH) 3 (ref. 23) and that depends on particle size. 18 Surface depletion leads also to an exceptionally low value for the surface Gibbs free energy. 24 This thermodynamic property is an essential parameter in the nucleation of ferrihydrite. The exceptionally low value for the surface Gibbs free energy makes ferrihydrite the most insoluble and stable iron (hydr)oxide at the nanoscale. 24 In the present study, we will compare the properties of ferrihydrite (Fh) with engineered ferritin (Ftn), first focusing on the ion adsorption of both materials. Our hypothesis is that these will be very similar because both materials have basically the same mineral structure. 9 One of the challenges of the current study is to describe the oxyanion adsorption behavior of iron (hydr)oxide as a function of particle diameter for a range that is difficult to assess using the traditional preparation method of ferrihydrite. We like to explore the ion adsorption behavior for particles on the ultra-nanometer scale for a range between the size of Fe (hydr)oxide nuclei and that of freshly precipitated ferrihydrite. This capability can be used to study the formation of Fe (hydr)oxides in the presence and absence of oxyanions as well as ferritin. Recently, Fe (hydr)oxide particles of various sizes have been produced with stepwise loading of ferritin. 6 The particles at the lowest Fe/Ftn loading (100) are very small. These clusters may grow atom by atom as well as by mutual attachment followed by fusion. Similar processes may occur during the synthesis of Fh. 25 27 The latter material is not stable and will transform into goethite and hematite above a size of nearly 8 nm. 24 When Fe (hydr)oxide is prepared below ph 7 by oxidation of a FeĲII) solution, lepidocrocite can be an intermediate phase. 28 In this process, FeĲII) adsorption 29 may play a role. The synthesis of Fe-loaded ferritin can be influenced by the specific adsorption of ions. 6,10 Therefore, a study of the adsorption behavior of such Fe clusters may contribute to our understanding of the nucleation and formation of Fe (hydr)oxide in complex aquatic media in general. In Ftn systems, simultaneously loaded with Fe and phosphate, 10 the synthesized nanoparticles have a much higher P/Fe ratio and the particles are expected to be very small. 6 The challenge is to estimate the possible size of these Fe-phosphate nuclei using surface complexation modelling. The results can be compared to the size of critical nuclei formed in the absence of Ftn. For this, we may use recent data for the phosphate adsorption 30 of small Fe hydroxide clusters produced with a flash-fast polymerization of FeĲIII) atph7. Our analysis will start with developing a surface complexation model (SCM) for ferrihydrite that describes the interface

and quantifies the site densities as a function of size. The SCM will be parameterized using phosphate and arsenate adsorption data for Fh. In recent literature, PO 4 binding has been studied extensively using freeze-dried Fh material. 31 35 However, this drying process with forced aggregation may affect the mineral structure and its surface properties. 14,36 Measurements of the phosphate adsorption by freshly prepared Fh in the wet state are relatively rare, sometimes fragmentary, and these published experiments cover a limited range of conditions 30,37,38 in contrast to AsO 4 that has been widely studied for Fh in the wet state. 39 43 Therefore, we will determine for our model parameterization the adsorption of PO 4 for freshly prepared ferrihydrite that will cover a wide range of ph values, PO 4 loadings, and ionic strengths. For AsO 4,we will rely on literature data. Our goal is to analyze these data using surface complexation modeling according to the latest insights into the surface and mineral structure of this material 18 that will be further developed here. Next, the parameterized model will be used to gain insight into the process of oxide particle formation in the absence and presence of Ftn and its relation to oxyanion adsorption. We will explore how well our mechanistic approach for Fh can predict the adsorption behavior of Fe-loaded Ftn. We will analyze the variation in phosphate adsorption for systems with a variable Fe-to-ferritin ratio. The results will be compared to systems without ferritin. The advantage of our mechanistic approach is the ability to predict competitive and cooperative ion adsorption for a variety of solution conditions as can be found in practical systems where ferritin is intended to be applied 6 and where Fh may form in natural soil and aquatic environments. Surface structure and model approach Mineral and surface structure Ferrihydrite can be built from parallel layers of regular Fe octahedra (Fe1) that are connected by FeĲIII) ions present in either strongly distorted octahedra (Fe2) or in tetrahedra (Fe3). According to the surface depletion model, 18 these Fe2 and Fe3 polyhedra are relatively unstable at the surface when forming singly coordinated groups. This leads to surface depletion for both polyhedra. Due to the large surface-tovolume ratio, about half of the unstable polyhedra will be lacking in freshly prepared Fh material at an average size of about 2.3 nm. 18 In the critical nuclei from which these Fh particles have been formed, 24 even more than three quarters of these polyhedra will be absent. Fig. 1 gives an example of the structure of a surfacedepleted Fh particle. This particle can be constructed 18 in Crystalmaker starting with defining a nearly spherical particle from which all those Fe2 and Fe3 polyhedra that form singly coordinated surface groups are removed. Next, the chemical composition as well as other properties can be calculated. The neutral particle of Fig. 1 has 214 Fe ions, 496 oxygen ions, and 350 protons, being equivalent with the approximate Fig. 1 Polyhedral representation of a surface-depleted Fh particle with sheets of Fe1 octahedra (purple blue) connected by Fe2 and Fe3 polyhedra (light colored). Oxygens are given as red spheres. No hydrogens are shown. The particle has 121 singly coordinated surface groups that may form monodentate surface complexes with phosphate or arsenate, and part of these groups (65) can additionally form so-called binuclear double corner ( 2 C) complexes. composition Fe 10 O 14 (OH) 2 7H 2 O. The first part of the formula represents the chemical composition of the mineral core and the second part is the excess amount of H 2 O due to the large contribution of surface groups. The molar mass of this neutral particle (M nano = 94.9 g mol 1 Fe) follows from the chemical composition. The equivalent spherical diameter (d = 2.57 nm) can be calculated from the particle volume that is found from the number of oxygens in the particle and the lattice volume of Fh (ref. 20) that can be expressed per mole of oxygen (10.7 10 6 m 3 mol 1 O). These numbers also allow calculation of the mass density of this particle (ρ = 3.80 10 6 g m 3 ) and, finally, its specific surface area (A = 615 m 2 g 1 Fh). Surface groups and site densities At the surface of Fh, various types of surface groups can be found that differ in the number of Fe ions that coordinate to the surface oxygens. Singly, doubly, and triply coordinated surface groups can be distinguished. Singly coordinated surface groups, i.e. FeOH 1/2, may bind a proton forming FeOH +1/2 2 and this group is able to react with arsenate, phosphate, and other oxyanions via ligand exchange, releasing the coordinated water molecule. Doubly coordinated groups are present as Fe 2 OH 0. These groups remain generally uncharged, in contrast to triply coordinated surface oxygens that may carry a ph-dependent charge. For representative crystal faces of Fh, site densities (N s ) can be derived. 18 These values can be used to estimate for the particle as a whole the site densities. 18 Another approach for deriving such site densities is particle construction with counting of the number of the various types of surface

oxygens. The related site densities can be found by expressing these numbers per unit surface area, as will be done here. We have constructed series of particles and calculated for the singly coordinated surface groups the site density that is given in Fig. 2a as a function of the particle size. The total site density (squares) slightly varies but the average is close to N s,tot = 5.8 ± 0.3 nm 2. At the surface of Fh, the singly coordinated groups are present at the corners of the Fe1 octahedra. When present in Fig. 2 Total site densities (N s,tot ) of singly (a) and triply (b) coordinated surface groups for constructed Fh particles of various sizes. All singly coordinated surface groups (a) can form monodentate surface complexes (squares). A subset of these sites (spheres) is also able to form binuclear double corner complexes (N s,dc ). The mean site densities have been used in our CD modeling (Table 1). The total site density of triply coordinated groups (b) increases steadily with size that varies between about the size of a critical Fh nucleus and that of freshly precipitated two-line Fh. For the triply coordinated surface groups at two important idealized crystal faces, 18 the total site density is given by a horizontal dotted line. As discussed in the text, there is a large variation in proton affinity for the triply coordinated groups, making the effective site density of Fe 3 O(H) much lower than the total site density of these groups. pairs at two different octahedra, double corner complexes can be formed, i.e. formation of binuclear bidentate surface complexes of phosphate (FeO) 2 PO 2 and arsenate (FeO) 2 AsO 2. For the various constructed Fh particles of different size, the number of FeOH(H) groups that can form this type of surface complexes can be counted. The resulting site densities are given in Fig. 2a (spheres) as a function of the particle diameter of Fh. This site density is quite sensitive to the precise local surface structure of the constructed particles and therefore quite variable. The average site density of these groups is found to be about N s,dc = 2.8 ± 0.6 nm 2. The total number of triply coordinated surface groups is relatively high for two important representative crystal faces, but for small particles, this site density diminishes (Fig. 2b). Analysis of the reactivity of triply coordinated groups is problematic. 18 A large variation in the types of these groups exists. The oxygen charge of the triply coordinated surface groups, calculated with a bond valence approach, shows a large variability, suggesting a very large range of proton affinities. 18 If the proton affinities are very different, triply coordinated groups with a low and a high proton affinity may be solely present as Fe 3 O 1/2 and Fe 3 OH +1/2, respectively, and mutual charge compensation will occur, resulting in an apparently low site density for variable proton binding as has been found for, e.g. the (011) face of goethite. 44 One of the present problems is that for Fh, the proton affinity is highly uncertain for a variety of triply coordinated surface groups. Moreover, part of the triply coordinated surface groups are situated more internally, which may lead to the formation of internal H bonds in the mineral lattice. This may reduce the contribution of such groups to the development of a phdependent surface charge. Because of the mentioned uncertainties, we will evaluate the possible contribution of Fe 3 O(H) groups to the surface charge during our analysis of the adsorption behavior of Fh. Surface area and molar mass Nanoparticles have a size-dependent molar mass and mass density due to the variable contribution of surface groups. It is essential to account for this in calculating for instance the specific surface area and other properties that will be used later in the text. For Fh, an equivalent specific surface area A (m 2 g 1 ) can be derived assuming spherical particles with a diameter d (m) and a mass density ρ (g m 3 ) according to: The mass density ρ nano can be calculated with: 24 in which M core is the molar mass of the core with composition FeO 1.4 (OH) 0.2 (81.65 g mol 1 Fe), n O is the mean number of oxygen ions per metal ion (n O = 1.6), and V O is the volume (1) (2)

of the lattice expressed per mole of oxygen ions (10.7 10 6 m 3 mol 1 O). The excess water density due to the presence of surface groups is represented by N H2 O (12.6 10 6 mol m 2 ) 24 and the molar mass of water by M H2 O (18 g mol 1 ). The mass density ρ nano varies with the contribution of surface groups that increase the volume of a particle more than its mass. For particles with a diameter d between 1.5 and 5.5 nm, the calculated mass density (ρ nano ) ranges from 3.1 10 +6 to 4.3 10 +6 gm 3, respectively. The corresponding number of Fe ions in the particles (n Fe ) follows from: 24 in which N Av is Avogadro's number and M nano is the overall molar mass of the Fh particle which can be found from: 24 According to the above equations, the molar mass M nano will decrease from 114 to 87 g mol 1 Fe for particles that increase in size from d = 1.5 to 5.5 nm and the corresponding amount of Fe per particle increases from n Fe =30ton Fe = 2600. The specific surface area (A) varies for this size range between A = 1260 m 2 g 1 Fh and A = 250 m 2 g 1 Fh. Charge distribution model Oxide surfaces usually have a ph-dependent charge that is related to the excess binding of protons by surface groups. The primary surface charge creates an electrostatic field that is neutralized by counter and co-ions. The metal (hydr)oxide (3) (4) surface may bind ions that are influenced by the field. Electrostatic energy is involved that highly depends on the position of the adsorbing ion in this electrostatic field. Near the surface, the electrostatic potential changes dramatically over very short distances. Therefore, the various ligands of an adsorbed phosphate or arsenate ion will be exposed to different local electrostatic potentials. 45 For this reason, the charge of an ion cannot be reduced to a single point charge in the calculation of the contribution of the electrostatic energy to the overall binding energy. In the charge distribution (CD) model, 45 the charge of an adsorbed ion is distributed in the compact part of the electrical double layer (EDL), known as the inner Stern layer. This compact part of the EDL can be extended by a second Stern layer that separates the inner Stern layer from the head end of the diffuse double layer. 46 In the latter, the electrostatic field is gradually further neutralized by the diffuse pattern of counter and co-ions. This double layer structure is illustrated in Fig. S1. The present model with the charge distribution coefficients and the stoichiometry coefficients of the formation reactions has been summarized in Table 1. Formation of mononuclear monodentate and binuclear bidentate complexes is allowed. In the model, we account for the influence of type and concentration of background electrolyte ions on the oxyanion binding 47 by using appropriate ion pair formation constants (Table S1 ). The site densities of the singly coordinated surface groups are from Fig. 2a. In our approach, the charge distribution in the surface complexes of phosphate and arsenate will not be assessed by fitting. Instead, we will use charge distribution coefficients that have been and will be derived from polyhedral geometries optimized by molecular orbital (MO) calculations applying density functional theory (DFT). 48,49 Table 1 Table defining the phosphate and arsenate surface reactions. Ionic charge due to change in ion adsorption is attributed to the surface (Δz 0 ) and 1-plane (Δz 1 ); no charge is added to the 2-plane (Δz 2 = 0). The extended Stern layer model used has two capacitance values (C 1 =1.15Fm 2 and C 2 =0.9Fm 2 for 2LFh). The site densities for FeOHĲa) and FeOHĲb) are N sĳa) =3.0nm 2 and N sĳb) =2.8nm 2, respectively. In the ESI, a table is given defining the reactions describing the primary charge Species a Type FeOHĲa) 1/2 FeOHĲb) 1/2 Δz 0 Δz 1 Δz 2 H + PO 4 3 AsO 4 3 log K FeOĲa) 0.22 PO 2 OH 1.28 MH 1 0 +0.28 1.28 0 2 1 0 log K MH FeOĲa) 0.17 0.33 POĲOH) 2 MH 2 1 0 +0.33 b 0.33 b 0 3 1 0 log K MH2 FeOĲb) 0.22 PO 2 OH 1.28 MH 0 1 +0.28 1.28 0 2 1 0 log K MH FeOĲb) 0.17 0.33 POĲOH) 2 MH 2 0 1 +0.33 b 0.33 b 0 3 1 0 log K MH2 (FeOĲb)) 0.54 1.46 2 PO 2 B 0 2 +0.46 1.46 0 2 1 0 c log K B (FeOĲb)) 0.35 2 POOH 0.65 BH 0 2 +0.65 0.65 0 3 1 0 c log K BH FeOĲa) 0.20 AsO 2 OH 1.30 MH 1 0 +0.30 1.30 0 2 0 1 log K MH FeOĲa) 0.17 0.33 AsOĲOH) 2 MH 2 1 0 +0.33 b 0.33 b 0 3 0 1 log K MH2 FeOĲb) 0.20 AsO 2 OH 1.30 MH 0 1 +0.30 1.30 0 2 0 1 log K MH FeOĲb) 0.17 0.33 AsOĲOH) 2 MH 2 0 1 +0.33 b 0.33 b 0 3 0 1 log K MH2 (FeOĲb)) 0.53 1.47 2 AsO 2 B 0 2 +0.47 1.47 0 2 0 1 c log K B (FeOĲb)) 0.42 2 AsOOH 0.58 BH 0 2 +0.58 P 0.58 0 3 0 1 ρan sĳa) ρan sĳb) 1 d P 2 d P3 d H tot PO 4tot AsO 4tot c log K BH a FeOHĲa) can form only monodentate surface complexes while FeOHĲb) can form mono- as well as bidentate complexes. b Present study. The other CD values are from ref. 48 and 49. c Relative surface concentrations (θ) are used which requires recalculation of log K to fit in a scheme with concentrations in mol L 1. 45 d Sums for the electrostatic columns ( P ) have been defined as described previously. 45

MO/DFT computations For a hydrated, doubly protonated monodentate complex bound to a template of 2 Fe-octahedra as defined previously, 48 the geometry has been optimized with molecular orbital (MO) calculations applying density functional theory (DFT) using the unrestricted Becke Perdew (BP86) model. Pseudo potentials (LACVP+**) were used as defined in Spartan'06 of Wavefunction Inc. The calculated geometries (Fig. S2 and Tables S2 and S3 ) have been interpreted using the Brown bond valence approach 50 (eqn S1 ) The resulting charge distribution (CD) has been corrected for interfacial water dipole orientation 46 resulting in CD coefficients being Δz 0 = +0.33 and Δz 1 = 0.33 valence units (v.u.) for the reaction FeOH 1/2 + PO 3 4 (aq) + 3 H + (aq) FeO 0.17 PO(OH) 0.33 2 +H 2 O(l). For the reaction FeOH 1/2 + AsO 3 4 (aq) + 3 H + (aq) FeO 0.17 AsO(OH) 0.33 2 +H 2 O(l), the calculated coefficients are Δz 0 = +0.33 and Δz 1 = 0.33 v.u. (see the ESI ). Experimental For all adsorption experiments and the preparation of Fh, the chemicals (analytical grade) used were dissolved in pre-boiled double distilled (bidest) water and contact with air was minimized. Materials Ferrihydrite was produced by the instantaneous addition of 0.02 M NaOH to a 0.9 L solution of 3.50 mm FeĲNO 3 ) 3 in 0.01 M HNO 3 until ph 8.5 was reached. After centrifugation for 20 minutes, the precipitate was re-suspended in 0.01 M NaNO 3 to a typical final volume of 150 ml, resulting in 2 2.5 g of Fh per L at a molar mass of M nano = 94.8 g mol 1 Fe. After aging in a closed bottle at room temperature over 4 hours since the neutralization, the adsorption experiments started. Adsorption edges Systems with a volume of 50.0 or 60.0 ml were prepared for measuring phosphate adsorption edges in 0.010, 0.50, and 0.95 M NaNO 3 by using 2, 5, or 15 ml of the prepared Fh suspension (ph 8.5, aged 4 hours) and appropriated amounts of electrolyte solutions (either 0.01 and 0.5, or 2 M NaNO 3 ). After the addition of acid (0.01 M HNO 3 /0.01 M NaNO 3 )or base (0.01 M NaOH) and bidest water, finally phosphate solution was added leading to an initial PO 4 concentration of 0.277 mm in the 2 ml Fh systems with 0.010 and 0.95 mm NaNO 3. For the 5 and 15 ml Fh systems in 0.50 M NaNO 3, the initial PO 4 concentration was 2.45 and 0.78 mm, respectively. The reaction time was 20 or 21 hours. Adsorption isotherms In the case of the determination of the adsorption isotherms, the initially produced Fh suspension was brought to ph 4, 7, 9, or 11 using 0.01 M HNO 3 in 0.01 M NaNO 3 or using 0.01 M NaOH before aging. After aging in a closed bottle over a total of 4 hours since the initial Fh formation, Fh systems in 0.010 M NaNO 3 were prepared. Variable amounts of Fh suspension ( 4 40 ml) kept under N 2 atmosphere were pipetted and volumes of 1 10 ml of 0.0100 M NaH 2 PO 4 solution were added leading to an intended fraction of adsorbed ions larger than at least 50% in order to ensure an accurate determination of the adsorption from the difference between the initial and the final phosphate concentration. The ph was readjusted and the systems with a total volume of around 50.0 ml (measured gravimetrically) were allowed to react over 4 or 21 hours. Before sampling, the ph was readjusted if needed. The ph values before the last adjustment deviated generally less than ΔpH 0 0.1. After centrifugation at 4500g for 20 minutes using no brake, the carefully pipetted supernatants were analyzed for orthophosphate using an accurate in-home colorimetric molybdenum-blue method. The Fe content of Fh was measured by atomic absorption spectrometry (AAS) after dissolution in acid. Results and discussion Phosphate Fh interaction In Fig. 3a, the phosphate adsorption isotherms are given for Fh suspensions at ph 4, 7, 9, and 11. The Fh used was aged over 4 hours before phosphate was added. The phosphate adsorption has been studied for a reaction time of 4 hours (open symbols) and 21 hours (colored symbols). The specific surface area used for scaling (58 10 +3 m 2 mol 1 Fe (ref. 24)) is consistent with TEM observations as discussed in the ESI. The data show that equilibrium was reached within 4 hours. At ph 4, the phosphate loading was very high, close to 4 μmol m 2. This loading is much higher than that found for well-structured goethite, where under the same conditions, the loading is close to 2.5 μmol m 2. 45 This is related to the much higher site density of Fh. The phosphate loading can also be expressed per mole of Fe (second y-axis) showing a maximum PO 4 /Fe ratio close to about 0.25 at ph 4. The phosphate adsorption has also been studied by making adsorption edges. In Fig. 3b, the effect of the background electrolyte concentration is shown. A small or no effect is observed at low ph (ph 4). Here, the particles are close to the isoelectric point (IEP) where no diffuse double layer is present. At higher ph values, the particles with adsorbed phosphate are negatively charged and a decrease in the extent of the DDL by the addition of background electrolyte (NaNO 3 ) diminishes the electrostatic field that is repulsive for the binding of phosphate at these ph values. It results in an increase in the phosphate adsorption. Note that at high ph, some data points of the 1 M level strongly deviate and in fact match those of the 0.01 M treatment. In Fig. 3c, additional experimental data are shown for two Fh systems with different Fh concentrations in 0.5 M NaNO 3 (triangles). For comparison, the 0.01 M data of Fig. 3b are given.

View Article Online Modelling phosphate adsorption Reactions. Our modeling of the phosphate adsorption started with using the same type of surface species as found for the goethite (α-feooh),48 namely formation of a binuclear bidentate surface complex ( (FeO)2PO2) and a protonated monodentate complex ( FeOPO2OH), respectively, defined with: (5) (6) The monodentate species was allowed to bind to both types of singly coordinated surface groups, being FeOH(a) and FeOH(b) (Table 1). The parameter evaluations were based on the measured percentage of adsorbed phosphate. Using in the modelling only a bidentate (B) and protonated monodentate (MH) species was not successful (R 2 = 0.90), particularly below ph 7, where the calculated adsorption remained too low. Therefore, protonation of the bidentate complex was introduced, according to: (7) The description of the adsorption improved (R 2 = 0.94), but the quality of the fit was still relatively low, particularly at a high PO4 concentration and acid conditions. By introducing additionally a doubly protonated monodentate species (MH2): (8) Fig. 3 Phosphate adsorption for 2LFh that was aged over 4 hours before the addition of PO4. The lines have been calculated using the CD model (Tables 1 and 2). The surface area is set to A = 611 m2 g 1 Fh at a molar mass of Mnano = 94.8 g mol 1 Fe (=58 10+3 m2 mol 1 Fe (ref. 24)). The adsorption isotherms in 0.010 M NaNO3 (a) were measured using variable concentrations of Fh ( 0.05 2 g Fh per L) to ensure a PO4 adsorption of at least 50%, which allows an accurate determination of the PO4 adsorption. Open symbols refer to a reaction time of 4 hours for PO4 and closed symbols are for 21 hours. (b) Phosphate adsorption edge in 0.010 M and 0.95 NaNO3 as a function of ph for a system with 0.0707 g Fh per L and an initial PO4 concentration of 0.277 mm. (c) Additional phosphate adsorption edges in 0.50 M NaNO3, for systems with an initial PO4 concentration of 0.78 mm containing 0.602 g Fh per L (blue triangles) and with an initial PO4 concentration of 2.45 mm having 0.201 g Fh per L (open triangles). The squares are for the 0.010 M NaNO3 system of Fig. 3b. a very good description (R 2 = 0.982) was found. Double protonation of a monodentate complex (MH2) has also been suggested by Tiberg et al.,37 measuring 4 adsorption edges for Fh (Fig. S3 ) covering a smaller range of conditions than that in our experiments. These authors did not use in their CD modeling a singly protonated monodentate complex (MH), but instead they used a supposed MH2 complex for which the charge distribution coefficients were fitted. However, the CD values obtained for MH2 (Δz0 = 0.5, Δz1 = 0.5 v.u.) deviate strongly from the CD values (Δz0 = 0.33, Δz1 = 0.33 v.u.), calculated from the presently MO/DFT optimized geometry of the hydrated MH2 complex (Fig. S2 ). With our MO/DFT derived parameters, the PO4 adsorption data of Tiberg et al.37 can also be described very well (Fig.

S3 ). In the final evaluation of the parameter values (Table 2), this data set has been included. Triply coordinated surface groups. In our modeling, the influence of triply coordinated surface groups has been evaluated. Triply coordinated surface groups introduce an additional surface charge that may affect indirectly the binding of PO 4 via electrostatic interactions. Excluding Fe 3 O(H) sites reduces the quality of the description of the phosphate adsorption, in particular the salt dependency near the isoelectric point (IEP) at a high PO 4 loading (Fig. 3b). As mentioned above, the triply coordinated surface groups largely vary in proton affinity. The surface charge, introduced by sites with a low proton affinity ( Fe 3 O 1/2 ), will be compensated for by high affinity sites ( Fe 3 OH +1/2 ), leading effectively to a lower reactive site density. 44 Our modeling suggests the presence of a relatively small amount of additional charge that is active in the phosphate adsorption process. If represented by an equivalent surface site with the same proton affinity as the singly coordinated surface groups (log K H = 8.1), the effective site density derived by fitting is N s ( Fe 3 O) = 1.4 ± 0.5 nm 2. This effective site density is low compared to the relatively high total number of Fe 3 O sites that is found at constructing 2LFh particles (Fig. 2b). The low value of the effective site density found by our modeling supports the hypothesis of mutual charge compensation due to a large variation in proton affinity of the sites. Modelling arsenate adsorption The adsorption of arsenate by freshly prepared Fh has been measured by many authors. We compiled a consistent subset of data (Fig. S4 S12 ) comprising 280 data points with a large range of variation in ph, initial AsO 4 concentration, type and concentration of the background electrolyte, and Fh concentration, collected by 6 different research groups. 39 43,51 Using Table 2 Log K values fitted for the adsorption reactions of PO 4 with Fh (R 2 = 0.983, RMSE = 3.84%, n = 203) and AsO 4 with Fh (R 2 =0.97,RMSE = 5.36%, n = 280). The surface groups of Fh comprise FeOHĲa), FeOHĲb), and Fe 3 O sites with a site density of 3.0, 2.8, and 1.4 ± 0.5 nm 2, respectively (see the text). For 2LFh with A =611m 2 g 1 Fh and M nano =94.8gmol 1 Fe, the capacitance values are C 1 =1.15Fm 2 and C 2 =0.9Fm 2 (see the text) Species a Type log K b FeO 0.22 PO 2 OH 1.28 MH log K MH = 26.36 ± 0.20 FeO 0.17 0.33 POĲOH) 2 MH 2 log K MH2 = 29.84 ± 0.23 (FeO) 0.54 1.46 2 PO 2 B log K B = 28.31 ± 0.04 (FeO) 0.35 2 POOH 0.65 BH log K BH = 33.52 ± 0.13 FeO 0.20 AsO 2 OH 1.30 MH log K MH = 23.88 ± 0.93 FeO 0.17 0.33 AsOĲOH) 2 MH 2 log K MH2 = 29.94 ± 0.06 (FeO) 0.53 1.47 2 AsO 2 B log K B = 27.70 ± 0.03 (FeO) 0.42 2 AsOOH 0.58 BH log K BH = 34.00 ± 0.09 a Bidentate complexes formed by interaction with singly coordinated surface groups of the type FeOHĲb). b Log K values for the monodentate species, formed from either FeOHĲa) or FeOHĲb), are kept equal. The proton affinity of MH and B complexes is discussed in the text. a combination of monodentate and bidentate surface complexes, 52 we are able to describe the data well. Evaluation of the percentage adsorbed shows a standard error of 5%. A good fit requires the species B, BH, MH, and MH 2. The parameter values and uncertainties are given in Table 2. Surface speciation The proton affinity constants of the B and MH surface complexes of both oxyanions can be derived by taking the Δlog K value of the formation constants of the related species. For PO 4, the proton affinity constants are log K H (B) = 5.2 ± 0.1 and log K H (MH) = 3.5 ± 0.3, respectively. The protonation constants (log K H ) of the AsO 4 surface species are higher, namely log K H (B) = 6.1 ± 0.9 and log K H (MH) = 6.3 ± 0.1, respectively. The relatively strong protonation of AsO 4 surface complexes may agree with our previous findings for a well-crystalized needle-shaped goethite, where we did resolve the intrinsic protonation constant of the bidentate AsO 4 complex (log K H = 3.7 ± 0.2), 49,53 but not that of the bidentate PO 4 complex. 48 The surface speciation of Fh is a function of ph but depends also on the PO 4 loading (Fig. 4). An increase in the oxyanion loading leads to the introduction of more negative charges in the interface, which stimulates electrostatically the H + binding to the outer ligands of the surface species. For Fh, protonation of the surface complexes is also enhanced because of the relatively high site density. It results in more anion adsorption and negative charge per m 2 facilitating proton binding for the same electrostatic reason. More generally, surface protonation may vary with crystal habit because of crystal faces that strongly differ in site density. For example, well-crystallized goethite has a needleshaped morphology with main crystal faces that have a low site density while the crystal faces at the top end have a higher site density of the same order as Fh (N s = 7 8nm 2 ). 20,54 The faces at the top ends may have more protonated surface complexes 55 and the overall adsorption behavior of goethite 45,56,57 will be determined by the relative contribution of both types of crystal faces. Phosphate interaction with ferritin In recent literature, 6,10 it has been shown that phosphate may strongly interact with Pyrococcus furiosus ferritin (Pf Ftn) loaded with Fe (hydr)oxide. Two types of experiments have been done, namely the measurement of adsorption isotherms at ph 7 and the measurement of the phosphate binding as a function of the Fe-loading of Pf Ftn. Adsorption isotherm (ph 7). The adsorption isotherms of phosphate have been collected by Jacobs et al. 10 and Sevcenco et al., 6 using Pf Ftn preloaded with 1000 and 1500 Fe per Pf Ftn molecule, respectively. The adsorption was measured at ph 7 using apo-ferritin without Fe as blank measurement. The adsorption can be scaled with the expected specific surface areas using eqn (1) (4). The experiments show a relatively large variation (Fig. 5) that may be due to the complexity of preparing well-loaded Pf Ftn and measuring very

View Article Online Fig. 5 Adsorption isotherm of phosphate binding to Pf ferritin at ph 7. The triangles and squares refer to two different experiments with 1000 Fe per PfFtn (ref. 10) (d 4.0 nm), and the spheres are for a recent experiment with a Fe/PfFtn ratio of 1500 (ref. 6) (d 4.6 nm). Details on the rescaling of the original data are given in the ESI. The line in Fig. 5 has been calculated with the CD model using the parameters of Table 2 with C1 = 1.04 F m 2 and C2 = 0.85 F m 2. The Na+ ions of the ph buffer ( 0.03 M) are considered as the counter ions in the Ftn cavity. low phosphate concentrations. Chemical heterogeneity may also play a role. The line in Fig. 5 has been calculated using the CD model. The agreement between data and model is just reasonable. Effect of Fe loading of Ftn. Sevcenco et al.6 performed an appealing experiment in which the effect of Fe loading on the phosphate binding per iron was studied at a relatively high phosphate concentration (Fig. 6). Equilibrium was reached within about 2 hours (Fig. S13 ). The Fe loading per Pf Ftn was varied by a factor of 20. In Fig. 6, the dotted line shows the model prediction assuming the formation of a single particle in each Pf Ftn cavity. The dashed line refers to the formation of two particles in each Pf Ftn cavity and the solid line is for three particles per cavity. The modeling will be discussed in detail in the next two sections. Modeling using a spherical double layer Fig. 4 Calculated ph-dependent surface speciation of phosphate as a function of the solution concentration in 0.01 M NaNO3. (a) At ph 9, the non-protonated bidentate (B) surface species is dominant and only a little amount of the mono-protonated monodentate complex (MH) is present. (b) At ph 7, the MH and BH surface species start to contribute. (c) At ph 4, protonated monodentate species (MH and MH2) become increasingly important. At an increase in loading, the MH species gradually converts into the MH2 surface species in agreement with the electrostatic reasoning (see the text). This also explains the decrease in the bidentate (B) surface species. The surface speciation qualitatively agrees with spectroscopic information.35,54 The lines in Fig. 6 are predictions of the amount of adsorbed PO4 per mole Fe added. The calculations have been done based on excess Fe loading accounting for the binding of 48 Fe ions to the metal-reactive centers of the 24-mer of Pf Ftn.2 Nevertheless, the calculated model results have been expressed per mole of added Fe, which allows direct comparison with the experimental information. At the lowest addition of Fe, the particles are extremely small and the surface curvature becomes very large. Therefore, we have applied spherical double layer theory for the compact part of the double layer. The capacitance of a spherical capacitor Cr with

Fig. 6 Phosphate binding per Fe added as a function of the amount of Fe added per Pf ferritin. The lines are predictions assuming the presence of a single Fe (hydr)oxide particle per PfFtn cavity (dotted line) or nucleation with the formation of two or three particles per PfFtn cavity (dashed and solid lines). The curves have been calculated using size-dependent capacitance values (eqn (9)) and mass densities (eqn (1) (4)), and only excess Fe is considered in the model calculations assuming that 48 Fe ions remain in the reactive centers of the 24-mer protein. 2 Data of Sevcenco et al. 6 radius r and thickness Δr of the Stern layer is related to the capacitance C of a flat plate capacitor according to: For the inner Stern layer, a thickness of Δr = 0.35 nm was used and for the outer layer Δr = 0.4 nm. 46 In combination with capacitance values for a flat plate, namely C 1 = 0.9 F m 2 and C 2 = 0.74 F m 2, 46 the equivalent capacitance values for the spherical capacitors can be calculated. For 2LFh (d = 2.6 nm), it results in C 1 1.15 F m 2 and C 2 0.9 F m 2. For an ultra-small particle (r 0.75 nm), the capacitance values will be approximately C 1 = 1.3 F m 2 and C 2 = 1.0 F m 2. Despite the introduction of these details, the prediction of the adsorption is far too low at the lowest additions of Fe, if formation of a single particle in the Ftn cavity is assumed (dotted line in Fig. 6). At low additions of Fe, the experimental phosphate binding is much higher, which may point to the presence of smaller particles produced during the initial nucleation. Nucleation processes Pristine particle formation of ferrihydrite. The size of Fh nuclei in the absence of Ftn can be estimated from important data concerning the instantaneous formation and growth of Fh nuclei at neutral ph as recently studied by Mao et al. 30 The mean particle size of these materials can be deduced from the measured phosphate adsorption, applying our parameterized CD model. The adsorption equilibrium was reached within less than 1 hour (Fig. S13 ). The collected (9) data for the phosphate adsorption as a function of ph and the time of aging of the initial Fh (ref. 30) can be described well as shown in Fig. S14a and b. The specific surface area A (m 2 per mole Fe) has been fitted with the CD model and from this result the average particle size (d) and number of Fe per particle (n Fe ) can be calculated (eqn (1) (4)). In the approach, we have accounted for the size-dependent variation of the molar mass and capacitance values (Table S4 ). For the Fh preparation at ph 7, the result is given in Fig. 7. The ph dependency is given in Fig. S15. Fig. 7 shows an excellent linear relationship between the mean number of Fe per particle and the time since neutralization. At zero time, the particles of the suspension contain on average about n Fe = 62 ± 3 at the conditions applied by Mao et al. 30 The corresponding mean particle size is d = 1.82 ± 0.03 nm and the specific surface area is A 970 ± 30 m 2 g 1. These numbers can be interpreted with the homogeneous nucleation theory using the recently derived surface Gibbs free energy 24 of Fh (γ = 0.186 ± 0.01 J m 2 ). The Gibbs free energy curve of particle formation and particle dissolution is a function of the particle size d and can be described with: 24 (10) in which γ is the surface Gibbs free energy (J m 2 ). The ratio Q so /K o so, expresses the super saturation of the solution relatively to virtual Fh bulk material of infinite ( ) size, being log Fig. 7 The time dependency of the average number of Fe present in engineered iron oxide nanoparticles initially produced by flash neutralization of an FeĲIII) solution at ph 7. The average number of Fe has been derived by fitting the specific surface areas (m 2 mol 1 Fe) in the description of the experimental PO 4 adsorption data of Mao et al. 30 using our CD model for ferrihydrite. In the modeling, we have accounted for the formation of a (FeOb) 2 CO surface complex 58,59 (see the ESI ). The linear relationship points to a constant growth/ dissolution rate of 18 Fe per particle per hour during Ostwald ripening. As discussed in the ESI, the data are in good agreement with the properties of the 2LFh produced for our experiments.

Fig. 8 Calculated (eqn (10)) change in Gibbs free energy ΔG R (aj = 10 18 J) at the formation or dissolution of a single Fh nanoparticle as a function of the particle size, accounting for the size dependency of the mass density ρ and molar mass M. The pathway of 2LFh (dark-blue line) has been calculated for an open system with a solution saturation of log Q so = 38.3 being equal to the solubility of freshly prepared 2LFh (d = 2.3 nm) without aging. The light-blue line is the reaction pathway during instantaneous neutralization of FeĲIII) producing smaller particles (d = 1.82 nm) with a higher solubility (log Q so = 37.3). At this solution condition, the critical size matches well with the smallest imaginable Fh-like cluster having a Keggin structure (n Fe = 13). The same cluster size can also be produced at the condition log Q so = 38.3 but only in the presence of a fictive amount of adsorbed oxyanions that decrease the surface Gibbs free energy by ΔG surf 0.05 Jm 2 (black line). The Gibbs free energy curves can also be given as a function of the number of Fe in the particle (Fig. S16 ). K so, = 40.6 ± 0.2. 24 In the above equation, the molar density ρ and the molar mass M are not constant but will depend on the diameter d. The above equation can be used to evaluate the reaction pathways at various solution conditions (Fig. 8). Based on a thermochemical analysis, 24 it has been suggested that in a 2LFh suspension with a mean size of 2.3 nm, the critical nucleus typically has a size of nearly d = 1.5 nm and will contain approximately 30 ± 5 Fe. 24 Fig. 8 gives the Gibbs free energy change of formation as well as dissolution of Fh particles (dark blue line) at an imposed supersaturation that is equivalent with the solubility product of 2LFH, i.e. log Q so = 38.3. 24 Our present interpretation is that the smallest particles in a 2LFh suspension are the result of dissolution during Ostwald ripening rather than nucleation, because nuclei are initially probably smaller during neutralization of an acid FeĲIII) solution. In the 2LFh suspension formed, particles can only grow at the expense of other (smaller) particles that will spontaneously dissolve during aging once driven back to the critical size, as has been described in more detail elsewhere. 24 The reaction pathway of instantaneously produced Fh is likely to be different from the pathway during Ostwald ripening. On average, the initial particles are much smaller and have a specific surface area of A = 970 ± 30 m 2 g 1. The corresponding solubility product (Q so ) can be calculated from the intrinsic solubility of Fh (K so ) and the surface Gibbs free energy (γ) according to 2.3RT log (Q so /K so )=AMγ (ref. 24) yielding log Q so = 37.3 ± 0.2. This value of log Q so matches with the experimental solubility of Fh produced by a rapid titration of Fe(II)/Fe(III) solutions with a base. 60 The reaction pathway at this solution condition is given in Fig. 8 with the light blue line, showing that the maximum value of ΔG R is lower and at a smaller particle size. The Gibbs free energy curve of particle formation/dissolution (Fig. 8) has a maximum that represents the unstable equilibrium state of a critical nucleus. The corresponding diameter of a critical nucleus can be found from the derivative of eqn (10). In our approach, this number was calculated numerically because of the size-dependency of the molar mass and mass density in eqn (10). The Ostwald Freundlich equation 24 is not fully valid and cannot be applied for this reason. With our numerical approach, one finds a critical diameter of d c = 1.22 ± 0.04 nm with M c = 138 ± 6 g mol 1 Fe and ρ c = 2.72 ± 0.06 10 6 gm 3 using log Q so = 37.3 ± 0.2. Within the uncertainties, the corresponding number of Fe in the critical nucleus (n Fe = 11 ± 2) matches very well with the number of Fe in a Fe 13 Keggin structure (Fig. S16 ). This suggests that formation of Fh may very well proceed via such a moiety. Pristine particle formation at Fe loading of Ftn. The Fe loading experiment of Pf Ftn (Fig. 6) points to the formation of more than one Fe (hydr)oxide particle per Pf Ftn molecule. 6,7 At the lowest Fe addition in the Ftn experiment of Fig. 6, an excess of 52 Fe per cavity has been oxidized. The lines in Fig. 6 give the model predictions assuming the presence of one, two, and three Fe (hydr)oxide particles per Pf Ftn. The prediction of the PO 4 adsorption with initially the formation of a triplet is close to the experimentally observed ratio of adsorbed PO 4 per total added Fe. Each cluster in the cavity will have initially on average 17 ± 1 Fe. At the subsequent additions of Fe, these nuclei may grow. Above a total addition of about 1000 1500 Fe/Pf Ftn, a model description that assumes the presence of a single particle becomes slightly better. At high loading, fusion to a singlet is likely. The formation and growth of three separated particles will be restricted by the size of the Pf Ftn cage. In the case of an addition of 1500 Fe in total, each of the three particles would contain about 500 Fe with a corresponding size near d = 3.2 nm (eqn (3)). Three of these particles will hardly fit into a cavity of 8 nm considering that separated particles require additionally some accommodation for the compact part of the double layer ( 1.5 nm. At a high Fe loading, the clusters may fuse and intergrow forming a single particle. 25 Nucleation in the presence of oxyanions. The above analysis indicates that the size of the nuclei is strongly determined by the solution conditions, in particular the super saturation, i.e. log Q so /K so. Another factor is the presence of adsorbing ions that will change the surface Gibbs free energy G surf (J m 2 ) as expected from the Gibbs equation: G surf = γ + Γ G Ads (11)